On Small Folkman Graphs Arrowing $K_2$ or $K_3$

TL;DR

This paper establishes new bounds for Folkman numbers related to small graphs avoiding certain subgraphs, using innovative graph construction methods and computational techniques.

Contribution

It provides novel bounds for Folkman numbers with parameters 2 and 3, introduces new theoretical insights, and demonstrates the effectiveness of semi-polycirculant graph generation.

Findings

Proved the existence of F_e(3,3;W_5)

Identified only one open case for F_e(3,3;H) with H =  P_2  P_3

Showcased the efficacy of semi-polycirculant graphs in finding Folkman graphs.

Abstract

For a graph $G$ and integers $a_i \geq 1$, we say that $G \xrightarrow[]{} (a_1, \ldots, a_k)^v$ if in any $k$-coloring of $G$'s vertices there exists a monochromatic $a_i$-clique for some color $i \in \{1,\ldots,k\}$. $G \xrightarrow[]{} (a_1, \ldots, a_k)^e$ is defined similarly, but for edge colorings. The Folkman number $F_v(a_1, \ldots, a_k; H)$ is the smallest number of vertices for which an $H$-free graph arrowing $(a_1, \ldots, a_k)^v$ exists. $F_e(a_1, \ldots, a_k; H)$ is defined similarly for edge-arrowing. In this work, we present new bounds for Folkman numbers where $a_i \in \{2,3\}$ and $k \leq 4$, while avoiding $K_n$, $J_n$, for $n \in \{4,5,6\}$, where $K_n$ is the complete graph on $n$ vertices and $J_n$ is $K_n$ missing an edge. We also present results for $C_4$-free and $W_5$-free graphs, where $C_4$ is the cycle on four vertices and $W_5$ is the wheel graph on five vertices. Notably, we prove the existence of $F_e(3,3;W_5)$, leaving only one graph, $\overline{P_2 \cup P_3}$, on five vertices for which the existence problem of $F_e(3,3;H)$ remains open. We provide some theoretical results that should aid in uncovering the existence of $F_v(3,3; \overline{P_2 \cup P_3})$. Our new bounds are the result of a variety of methods involving filters, extension, semi-polycirculant graphs, locally linear graphs, and the modification of special graphs. Most of our bounds are from the semi-polycirculant graph generator, showcasing its efficacy for finding witness Folkman graphs.